Applied Electromagnetic Waves ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018 Notes 22 Antenna Patterns

Infinitesimal Dipole The infinitesimal dipole current element is shown below. The dipole moment (amplitude) is defined as Il. The infinitesimal dipole is the foundation for many practical wire antennas. From Maxwell’s equations we can calculate the fields radiated by this source (e.g., see Chapter 7 of the Shen and Kong textbook).

Infinitesimal Dipole (cont.) The exact fields of the infinitesimal dipole in spherical coordinates are

Infinitesimal Dipole (cont.) In the far field (r  ) we have: Hence, we can identify

Infinitesimal Dipole (cont.) The radiation pattern is shown below. -9 -3 -6 0 dB 30° 60° 120° 150° 45o HPBW = 90o

Infinitesimal Dipole (cont.) The directivity of the infinitesimal dipole is now calculated Hence

Infinitesimal Dipole (cont.) Evaluating the integrals, we have: Hence, we have

Infinitesimal Dipole (cont.) -9 -3 -6 0 dB 30° 60° 120° 150° The far-field pattern is shown, with the directivity labeled at two points.

Wire Antenna A center-fed wire antenna is shown below. Feed A good approximation to the current is:

Wire Antenna (cont.) A sketch of the current is shown below for two cases. Resonant dipole (l = 0 / 2, k0h =  / 2) Short dipole (l <<0) Use

Wire Antenna (cont.) Short Dipole The average value of the current is I0 / 2. Infinitesimal dipole: Short dipole (l <<0 / 2) Short dipole:

Wire Antenna (cont.) For an arbitrary length dipole wire antenna, we need to consider the radiation by each differential piece of the current. Far-field observation point Feed Infinitesimal dipole: Wire antenna:

Far-field observation point Wire Antenna (cont.) Far-field observation point Feed

Far-field observation point Wire Antenna (cont.) Far-field observation point Feed Note:

Far-field observation point Wire Antenna (cont.) Far-field observation point Feed It can be shown that this approximation is accurate when

Far-field observation point Wire Antenna (cont.) Far-field observation point Feed Hence we have:

Wire Antenna (cont.) We define the array factor of the wire antenna: We then have the following result for the far-field pattern of the wire antenna: Note: The term in front of the array factor is the far-field pattern of the unit-amplitude infinitesimal dipole.

Wire Antenna (cont.) Using our assumed approximate current function we have: Hence The result is (derivation omitted):

Wire Antenna (cont.) In summary, we have: Thus, we have:

Wire Antenna (cont.) For a resonant half-wave dipole antenna: or

Wire Antenna (cont.) The directivity is: The result (from numerical calculations) is:

Wire Antenna (cont.) Results

Wire Antenna (cont.) Radiated Power: Simplify using

Wire Antenna (cont.) Performing the  integral gives us After simplifying, the result is then

For a resonant antenna (l  0 / 2), Xin = 0. Wire Antenna (cont.) The radiation resistance is defined from Feed Circuit Model For a resonant antenna (l  0 / 2), Xin = 0.

The radiation resistance is now evaluated. Wire Antenna (cont.) The radiation resistance is now evaluated. Using the previous formula for Prad, we have: Resonant l0 / 2 Dipole:

The result can be extended to the case of a monopole antenna Wire Antenna (cont.) The result can be extended to the case of a monopole antenna Feeding coax (see the next slide)

This can be justified as shown. Wire Antenna (cont.) + - Monopole This can be justified as shown. + - Dipole Virtual ground